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Some remarks and clarifications concerning the restrictions placed on thermodynamic processes
International Journal of Engineering Science 140 (2019) 26–34
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Some remarks and clarifications concerning the restrictions placed on thermodynamic processes K.R. Rajagopal, A.R. Srinivasa∗ Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA
a r t i c l e
i n f o
Article history: Received 24 February 2019 Revised 25 April 2019 Accepted 26 April 2019
a b s t r a c t Various disparate restrictions are assumed with regard to bodies undergoing thermodynamic processes, in addition to the second law, to cull the class of constitutive relations for describing their response. In this paper we critically evaluate some of these additional restrictions. After a brief discussion of the various forms of the second law due to the pioneers in the field we turn our attention to an assessment of additional stringent restrictions due to Onsager, Prigogine, Ziegler, and Edelen. With the use of specific examples we show the pitfalls and inadequacies of these additional restrictions. The importance of identifying the “cause” (which we refer to as the “determinant” which is incorrectly referred to as “flux”) and its “effect” (which we refer to as the “resultant” which is incorrectly referred to as “affinity”) and not merely splitting the “determinant” on the basis of mathematical ease rather than understanding the underlying physics, is emphasized by considering specific examples. © 2019 Published by Elsevier Ltd.
1. Introduction This paper is devoted to a critical assessment of the validity and legitimacy of the disparate thermodynamic restrictions that are in vogue, that unfortunately do not speak with one voice. This dissonance is mostly a consequence of restrictions that are demanded in addition to the second law of thermodynamics, a law that is considered to be inviolate, which however is interpreted in a variety of ways. These additional restrictions are enforced to restrict the class of processes that a body can undergo as the second law allows for too large a class of processes to be possible, leading to a very large class of mathematical models from which one can make a choice for describing the response of bodies that one is interested in characterizing. While one might opine that having a very large class of constitutive relations to choose from could be viewed as being a very desirable situation, such models may exhibit other physically undesirable properties. Thus, it might be reasonable from the start to build in these additional requirements. Before we direct our attention to the confusion that is caused by the additional demands, it is necessary to discuss the various interpretations of the second law.The origins of the second law of thermodynamics can be traced to the seminal work of Carnot (1960) (Dover edition of the original which was published in 1824) entitled “Reflections on the Motive Power of Fire, and on Machines Fitted to Develop That Power”. Current versions of the second law cannot be easily recognized as stemming from that work, but nonetheless they do. There are numerous interpretations of the second law, starting with
∗
Corresponding author. E-mail address: [email protected] (A.R. Srinivasa).
https://doi.org/10.1016/j.ijengsci.2019.04.003 0020-7225/© 2019 Published by Elsevier Ltd.
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the following early statement due to Clausius1 (Clausius, 1976 also stated that: (1) The energy of the universe is constant, and (2) The entropy of the universe strives to attain a maximum), and Kelvin,2 and the inaccessibility statement due to Carathéodory (1909),3 the statement due to Planck (1949),4 the inequality that is referred to as the Clausius–Duhem inequality, etc. Irrespective of which statement is taken as the second law, the implications of all of them are the essentially the same, namely that heat cannot flow spontaneously from a colder to a hotter body and that the entropy of an isolated system cannot decrease or that the entropy of an isolated system tends to a maximum. The physicist Eddington Stachel (2016) bemoaned the inexorability and the cataclysmic consequences of the second law when he remarked “At present we can see no way in which an attack on the second law of thermodynamics could possibly succeed, and I confess personally I have no great desire that it should succeed in averting the final running-down of the universe.” While one comes across papers that discuss the possibility of the violation of the second law, supposedly at very small scales, these results are far from convincing and are yet to be accepted by the scientific community at large. This paper mainly considers the more stringent restrictions on thermodynamic processes than that required by the second law of thermodynamics. However, before getting into a discussion of these more exacting restrictions, a few remarks on how the second law itself is used as a constraint on thermodynamic processes of continua are in order. The second law as it stands concerns an isolated system and is a global statement. In continuum mechanics however the second law is assumed to hold for all arbitrary sub-parts of the body, which in the instance of all the variables in question being continuous leads to a local form of the law, and the second law is then enforced in the local form. This is a much more strict stipulation than the global form of the second law which allows the entropy in certain sub-parts within an isolated system to decrease as long as the entropy of the isolated system as a whole increases. One also finds the second law being used to enforce restrictions of thermodynamic processes associated with open systems which is clearly not the intent of the second law. One could argue that there are truly no closed systems that are sub-parts of our universe, every sub-part being in some way influenced by its environment and that the second law is an idealization and even as an idealization it might only be applicable to the Universe as a whole. Of course, the idealization that one could have such isolated bodies has stood us well with regard to explanations of processes in nature and as Eddington has observed, to date no such violation has been reported. Even if one accepts the use of the second law in its local form as a proper restriction on thermodynamic processes that a body undergoes, there is an additional requirement that is commonly demanded in continuum thermodynamics that is definitely not warranted, namely assuming that the body in question can undergo arbitrary thermodynamic processes, the second law in its local form (or to be more more precise, the constitutive relation that is used to describe the body), holding in all these arbitrary processes. The problem is a body cannot be described by the same constitutive relation in all arbitrary thermodynamic processes. In fact, the nature of a body is determined by the class of thermodynamic processes that it can undergo. Put differently, the class of constitutive relations that one assigns to a body is valid only for a certain class of processes. For instance, in the case of sufficiently small deformations that take place slowly, it might be reasonable to describe rubber as an elastic body. However, if the process is sufficiently fast, rubber can crystallize and one can no more consider is as elastic. Thus, in studying the response of a body to external stimuli, we need to know apriori the class of processes that the body can experience and find sufficient conditions, which might imply the class of constitutive relations that are appropriate, in order for the second law to be met. An even more telling example is that of a solid that melts. As a solid, it might have been described through constitutive relations for an elastic body and/or an inelastic body; as a liquid it would have a totally different constitutive relation. All that being said, let us assume that we can indeed enforce the second law of thermodynamics locally, dispense with our discussion of the second law and, and consider some of the stricter local restrictions that are demanded of thermodynamic processes. Onsager (1931)(see also Glansdorff and Prigogine (1972) for a discussion of the ideas of Onasager and the ensuing developments) using the assumption of microscopic reversibility, built upon the work of Lord Rayleigh (1873), and proposed a reciprocity principle that thermodynamic processes have to satisfy. He required that the rate of dissipation be minimized and showed that in an isolated body the reciprocity principle was equivalent to a variational statement that “the rate of increase in entropy, less the dissipation function, be a maximum”. Onsager’s reciprocity principle leads to linear laws such as Fourier’s law, Fick’s law, etc., and thus was hailed as a great advance in thermodynamics. Unfortunately, the scope of Onsager’s work is quite restrictive. Onsager’s reciprocal relations, and theorems related to it, only apply when the rate of entropy production is a quadratic form. The relations do not hold for yield-like phenomena exhibited by inelastic bodies and for most nonlinear response exhibited by bodies.
1 Clausius’ statement of the second law in Clausius (1862) as paraphrased by Thomson reads: “It is impossible for a self-acting machine, unaided by any external agency, to convey heat form one body to another at a higher temperature” 2 Kelvin’s statement of the second law [6] : “It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects”. 3 Caratheodory’s statement of the second law (Carathéodory, 1909): “In every arbitrarily close neighborhood of a given initial state there exists states which cannot be approached arbitrarily closely by adiabatic processes.” 4 Planck’s statement of the second law (Planck, 1949): “It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.”
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Ziegler (1963), Ziegler (1972) and Ziegler and Wehrli (1987) proposed two different “principles” wherein he required that rate of dissipation, expressed in terms of “Fluxes” and “Affinities” has to be orthogonal to the level surfaces of a dissipation function or that the rate of dissipation functions should be maximized. Rajagopal and Srinivasa (2008) also required that the rate of dissipation be maximized. However, as pointed out by them, there is a fundamental difference between the ideas of Ziegler (1963) and those of Rajagopal and Srinivasa (2008), in particular in the quantities that can be varied in the maximization process. In his work in 1963, Ziegler (1963) makes the following statements: “In a given stage of the deformation process, the entropy production inside the macrosystem is therefore a function of the (velocities) alone. So is the (rate of dissipation function) which is thus defined without recourse to the irreversible forces”. This statement is not true in general. The state variables that go into the determination of the rate of entropy production (rate of dissipation) are not merely velocities. Ziegler also makes the statement that (see page 134 line 3): “If the irreversible Force is prescribed, the actual quasistatic velocity maximizes the rate of dissipation”. That is, the maximization is achieved by keeping the force fixed and varying the velocities under the further assumption that the process is quasistatic. The focus on varying the velocities as opposed to varying the forces has deep philosophical implications related to the state of a dynamical system.In determining what quantities ought to be varied, one has to consider the appropriate dynamical system and determine its state variables so that “causes” and “effects” can be properly distinguished. This point is missed when considering classical thermodynamics since it is mostly concerned with quasistatic processes and cause and effect can be interchanged with suitable Legendre transformations. The use of the “generalized velocities” in cases where the dynamical system has inertia (so that the forces affect the accelerations) is philosophically problematic since, for such systems the position and velocity are both considered as part of description of the current state of the system and hence cannot be varied. What can be varied is the driving forces in these cases, since that is the cause for the change in the system. The situation is quite different for cases without inertia (for example with the plastic strain rate in classical plasticity) Ziegler’s approach, which focuses on generalized velocities alone will not lead to the determination of the equations that describe numerous phenomena. It will not lead to many of the implicit models that are used in mechanics. Practical utility aside, philosophically such an approach is wrong headed as it implies that the causes (namely forces) are kept constant and the velocities (effects) are maximized to obtain the results. Causality has been turned topsy-turvy! On the other hand, Rajagopal and Srinivasa (20 05, 20 08) require that “given the state variables of the dynamical system, the irreversible forces are such that the rate of dissipation is maximized”. That is, the effects (rate of change of the state variables) are the result of the causes (forces) which are changed to achieve the maximization. Ziegler’s approach can only lead to an expression for the stress in terms of the state variables and cannot yield implicit relationships between the forces and the state variables, this is clearly philosophically unsound as explained in detail by Rajagopal (2011). The shortcomings of Ziegler’s approach have been discussed in detail by Rajagopal and Srinivasa (2005) and Rajagopal and Srinivasa (2008). Other variational principles have been proposed (see. e.g., Edelen, 1973, Goddard (2014)) which have been adapted to study dissipative systems, but such approaches can be only used to study static problems since they explicitly ignore inertia and so they are inapplicable in dynamic problems(an example of the same is the recent work by Goddard (2014)). Such studies while applicable to the very special situation being considered might give the erroneous impression that they are valid when dynamics is to be taken into account. Given that one of the most important issues concerning the state of the body being its “stability”, dynamics cannot be ignored in most situations of interest. Furthermore, if one is truly interested in “thermodynamics” and not “thermostatics”, employing ideas that fail whenever dynamical effects come into play is clearly inappropriate. Rajagopal and Srinivasa (2005) have demonstrated how the ideas of Onsager could be made compatible with more general forms for the rate of entropy production and the requirement that the rate of entropy production be maximized without restrictions to quasistatic processes. There have been claims made in recent papers (see Goddard, 2014) that the proper way to establish constitutive relations for dissipative processes is to use the decomposition developed and used by Edelen (1973). In this paper we show that there is nothing particularly new to be gained by such an approach (since in these cases, it amounts to nothing more than a definition of a suitable vector and an explicit requirement that the terms orthogonal to it should be set to zero without any particular physical basis) and more importantly it cannot be applied to several situations where the requirement of the rate of entropy production should be maximum can be imposed, such as the dynamic problems mentioned earlier. Before getting into the main body of the work we would like to discuss a couple of issues that might seem nitpicky but which we think needs to be dealt with. The first is the poor choice of terminology that could confuse issues being discussed. In most of the papers devoted to irreversible thermodynamics, we come across the terms “fluxes” and “affinities” that are appended to certain variables. These terminologies are inappropriate to describe these quantities. What we have are quantities that can be identified as causes and those that can be identified as effects. For instance, consider the quantity that we refer to as stress. This is incorrectly labeled as being a flux. The word flux as a noun means flowing or flow and there is nothing about stress within the context of continuum mechanics (especially in solids) to think of it as flowing. The stress, which is a consequence of the contact force, causes deformation. As Truesdell (2012) remarks: “In continuum mechanics the forces of interest are contact forces, which are specified by the stress tensor.”; hence we shall refer to it as a “determinant”, an adjective that means that which determines the effect. The gradient of velocity in these works is identified as being the “Affinity” that corresponds to the stress. The word “Affinity means (Simpson & Weiner, 20 0 0) “an attractive force between substances and particles that causes them to enter into and remain in chemical combination”. There is nothing with regard
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to the nature of the velocity gradient to think of it as an “Affinity” in the sense of the above definition of an attractive force. If by “Affinity” one means “closely connected”, one cannot see any connection between the stress and the velocity gradient other than one being a cause and the other its effect. Thus, it would then make eminent sense to refer to the stress as a “determinant” and the velocity gradient as the “resultant”, that which is a result of the cause. Thus, we shall refer to quantities that cause to belong to the set of “determinants” and the effects to belong to the set of “resultants”. The second issue is the role of temperature, whether it is a “determinant” or a “resultant”. One comes across the following motivation/definition of temperature. According to the Simpson and Weiner (20 0 0), temperature is defined as “The state of a substance or body with regard to sensible warmth or coldness, referred to some standard of comparison, that quality or condition of a body which in degree varies directly with the amount of heat contained in the body”. And according to Maxwell (1891): “The temperature of a body is its thermal state considered with reference to the power of communicating heat to other bodies”. In both the above definitions, we notice that in order to define “temperature” the concept is based on first comprehending what is meant by “transfer of heat” as it is in the context of being able to transfer of heat that one understands the notion of “temperature”. Fosdick and Rajagopal (1983) have shown that by assuming the notion of the “transfer of heat” (transfer of thermal energy) one can prove that there exists a one-dimensional manifold that one can associate with the notion of “temperature”. Put differently, the concept of the notion of “transfer of heat” leads to the notion of “temperature”, thus we shall identify the “transfer of heat” as a “determinant” and the gradient of temperature as a “resultant”. The arrangement of the paper is as follows. After an introduction to the mathematical and thermodynamic preliminaries in Section 2, we discuss the procedures adopted for the determination of an entropy production (dissipation) potential and the status of Onsager’s reciprocity relation Onsager (1931), the work of Ziegler (1963), and the use of the ideas of Edelen (1973) by others Goddard (2014) and the consequences of requiring that such a choice be consistent with the demands of causality. We discuss the fact that Ziegler’s approach turns causality on its head and also does not allow one to obtain the cls of response functions that are necessary to describe a plethora of natural phenomena. Section 4 is devoted to a discussion of the procedure applied to the Reiner–Rivlin fluids and elasto-plastic solids. In the final section, we offer some cautionary remarks concerning variational generalizations of the second law. 2. Preliminaries Consider a body B that is made up of particles P that occupy positions X in an initial configuration κ r and x in the current configuration κ . The motion χ, the velocity v and the deformation gradient F are given by the usual definitions
x = χ ( X, t ),
v = x˙ ,
F=
∂x . ∂X
(1)
where the superposed dot represents the time derivative holding X fixed. For bodies that do not allow for diffusion, the motion is constrained by the conservation laws that take the form
˙ = −divv,
v˙ = divσ ,
u˙ = div(h ) + σ •L
(2)
where ϱ is the mass density, σ is the Cauchy stress and u is the specific internal energy, h is the heat flux and L := ∇ v is the velocity gradient. It is evident that these are in the form of a dynamical system whose evolution can be determined once constitutive relationships between the fluxes and the basic variables can be specified, so that the system can be closed. Specifically, once the initial conditions for x, v, ϱ and u are given, in principle the above Eqs. (1)b and (2 a,b,c) furnish sufficient equations for the evolution of the system. In determining the constitutive relations we have to begin with the fundamental equation of state which we introduce here in the form
η = ηˆ (F, , u ),
∂η 1 := ≥ 0 ∂u θ
(3)
which introduces the specific entropy η as a function of the internal energy and other kinematical variables, and whose evolution is governed by
η˙ = div(h/θ ) + ζ ,
ζ ≥ 0.
(4)
In the conventional approach to continuum thermodynamics, this additional equation is used as a constraint on the dynamical system, in the sense that the constitutive relations for the stress and the heat flux should be consistent with meeting the constraint for the class of processes under consideration. Here we caution the readers that the constraint ξ ≥ 0 is much more stringent than the statements of the second law of thermodynamics (which usually involve the whole body undergoing cycles of state). However, for our purpose we will retain the more stringent definition that we use here. The fundamental equation of state in the form (3) is too cumbersome to use and usually we use Legendre transforms to convert it to more convenient variables such as the Helmholtz or Gibbs Potentials. With the use of the Helmholtz potential, which can be written as ψ (F, ϱ, θ ), the consistency of the constitutive relations requires that η = −∂ψ /∂θ and
σ E •∇ v − h•∇θ = θ ζ , := ξ ≥ 0
(5)
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where the “extra stress” σ E is given by
∂ψ t 2 ∂ψ σ = σ − F + I , ∂F ∂ E
(6)
where the notation ( · )t represents the transpose operation. All that we need to do is to pick constitutive relations for σ and q such that (5) is non-negative for the process class under consideration. It is important to note that the statements of the second law of thermodynamics refer only to restrictions on the process classes (a typical statement being that due to Clausius (1862) that reads: “It is impossible for a self-acting machine, unaided by any external agency, to convey heat form one body to another at a higher temperature”.) and actually speaks nothing about constitutive relations. In keeping with this, if we choose any particular constitutive relation, then the non negativity of the rate of dissipation ξ will restrict the process class. 3. Further restrictions of the constitutive relations: Quest for a scalar potential for dissipative processes We note that Eq. (5) can be written as an inner product between the determinant variables J = {σ E , −h} and the resultants Q = {∇ v, ∇ θ } in the form
J •Q = ξ ≥ 0
(7)
The question is, what should we choose for Q? As with the case of hyperelastic response, there has been a concerted effort made to provide a rationale for obtaining Q in terms of a single scalar “potential”. Several geometrical as well as variational arguments have been made in the literature that are related to the dissipative response functions, the most notable of them being the Onsager reciprocity principle (which is based on the notion of microscopic reversibility) for linear constitutive relations and its related variational principle for steady state which is referred to as Prigogine’s principle of minimum rate of dissipation (Glansdorff & Prigogine, 1972). No such general result has been widely accepted in the literature with regard to non-linear dissipative response. The most widely used nonlinear dissipative response is provided by power law (or generalized Newtonian fluids) and elasto-plasticity. Formulation of the constitutive relations for the latter has traditionally pursued a geometric outlook using the notion of yield surfaces and loading and unloading directions. As has been extensively discussed by Rajagopal and Srinivasa (20 05, 20 08), Ziegler (1963, 1972) and Ziegler and Wehrli (1987) put forth two different, but related ideas based on the notion that not only the equation of state but a constitutive equation for the rate of dissipation in terms of the “resultants” Q needs to be provided and that (1) the “determinants” J should be such that the rate of dissipation be maximized. Later he modified it into a purely geometric criterion that the “determinants” should be along the normal to the surface of constant rate of dissipation. In a nutshell Ziegler (1972) approach can be stated in the form that the “determinants” J should be directed along the normal to level sets of the rate of dissipation function (Ziegler states this as being “orthogonal to the D-Surfaces”), i.e. since the normal to the level sets of ξ are given by ∂ ξ /∂ Q, Ziegler’s assumption implies that
ξ = ξˆ (Q ),
J=λ
∂ξ , ∂Q
(8)
the value of λ is determined by the condition (7). By substituting (8) into (7) and solving for λ we arrive at
λ=
ξ
(9)
Q•∂ξ /∂ Q
In a recent paper by Goddard (2014), this approach (and approaches related to this) was questioned—the argument being based on a purely mathematical decomposition of the flux vector J(q) (see Edelen, 1973). Specifically the approach advocated by Goddard (2014) is based on the following observation: Given any constitutive relation J(Q), (so that ξ (Q ) := J(Q )•(Q )) we can construct5 a function φ (Q) by setting
φ (q ) :=
1 0
1
λ
ξ (λQ ) dλ ⇒
∂φ • Q = ξ (Q ) ∂Q
(10)
In view of the above definition for φ , we can always decompose the vector J as
J=
∂φ + U, ∂Q
U •Q = 0
(11)
In other words, any “determinant” vector J which is a function of the “resultants” Q can be UNIQUELY written as a gradient of a scalar and a non-dissipative flux term U. They go on to posit that the non-dissipative term U be set to zero to describe purely dissipative systems.
5
The general result obtained by Edelen (1973) is vastly broader, but in the interest of clarity it is sufficient to discuss a more restricted version.
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Goddard points out cases where the results (11) (with U set to zero) and (8) are the same in many cases, and also that somehow (8) is incorrect since it is not generally in accordance with (11). He points to the case of Reiner–Rivlin fluids as an example. We show later that this claim is incorrect. As has been pointed out by Rajagopal and Srinivasa (20 05, 20 08), there are several issues with both the approaches (11) and (8) none of them having to do with the mathematics of the approaches but with their philosophy. We first emphasize that, irrespective of the philosopical issues, the assumption U = 0 in (11) is a completely ad-hoc constitutive choice that is not based on any physical notions but a purely mathematical assumption. On the other hand, (8) is based on a physical assumption. 3.1. Cause, effect and teleology Classical thermodynamics, unlike Quantum thermodynamics, is well anchored in determinism and causality plays a central role in its development. Delving deeply into the philosophical underpinning of classical thermodynamics is not necessary in making our case with regard to the issues on hand. A cursory understanding of the role of causality is sufficient for our purposes. In the particular case of dissipative response, from a purely mathematical point of view, one can establish the following general result: Given ξ (Q) and any scalar valued function φ (Q) whose level sets are star shaped with respect to the origin, it is always possible to write the “determinants” J as
J=λ
∂φ + U, ∂Q
U •Q = 0
(12)
The proof is trivial: simply set λ = ξ (Q )/(Q•∂φ /∂ Q ). The denominator is non zero owing to the star shaped nature of the level sets and the result follows. It is now possible to choose U = 0 and then for any function ξ we will have a constitutive relation in terms of a scalar function φ . In the context of plasticity, these two functions ξ and φ are related to the yield function and the loading function, respectively. The two special cases are when this function is ξ itself, in which case we get the orthogonality condition of Ziegler and when this function is defined by (10) we get (11). Again, within the context of plasticity, the identification of the yield and loading functions as being the same gives rise to the “normality rule”. Thus, mathematically there is nothing particularly unique in preferring Ziegler’s orthogonality (Ziegler & Wehrli, 1987) or (11), other than the requirement of specifying two functions instead of one. A similar situation is found with constraint responses, where we have the geometrical idea that the generalized constraint force is normal to the constraint surface. A search for a physical reason for this condition led to two ideas–that the constraints did no virtual work and further the “Gauss principle of least constraint” (Gauß (1829), See also O’Reilly and Srinivasa (2001) and Rajagopal and Srinivasa (2005) for applications in continua), which provides a physical basis for the assumed normality condition. This points to the need for seeking other physical explanations for why one might prefer one version versus the other. We propose to offer such an explanation here. 3.2. Cause and effect are reversed in the conventional approaches of Ziegler and Wehrli (1987) and Goddard (2014) We first note that an “orthogonality condition” such as (11) has no “meaning” other than geometric convenience which may be corroborated (or at least not completely disproved) by experimental evidence. In general for it to have physical meaning, such an orthorogonality condition should be connected to a statement of physical significance. For example, the “normality” rule of plasticity is connected with the notion of maximum plastic work or the idea of the work inequality both of which provide a physical justification for the geometrical idea. When we consider such an approach, we also need to consider: In order to describe an optimum principle, we need to know the parameters that are fixedby the current state of the system and the parameters that can be varied. We note that as a dynamical system given by (2), the causes that drive the system are the “determinants”, i.e. J is the cause. Moreover, at any instant of time Q is already specified. Thus, any maximum principle that is generally valid should consider the (as yet unspecified) “determinants” J as being varied and not the “resultants” Q. In terms of meaning, if we assume that ξ is a function of Q, its value is fixed at each instant of time (since the state is known). So it is not suitable to consider ξ as a function only of the state variables, it should be considered as a function of the as yet unspecified causal variables J so that its values can be varied by choosing different values for J. In other words for the dynamical system considered above, it is proper to consider a maximum dissipation assumption of the form: Given a rate of dissipation function ξ (J) among all the choices for J that satisfy (7) for a given Q), the actual values for J are such that the rate of dissipation is maximized. The above problem can be stated as a constrained maximization of a function f (λ, J ) = (ξ (J ) − λ(Q•J − ξ (J )) ) (using a Lagrange multiplier to enforce the constraint (7)
J∗ (Q ) = argmaxJ f (λ, J )
(13)
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Under suitable smoothness assumptions on ξ (J), so that the standard methods of calculus can be applied, the above equation implies that the resultants are normal to the constant dissipation surface in “determinant” space, i.e.,
∂f ∂ξ =0⇒Q=μ , ∂J ∂J
(14)
where μ = (1 + λ )/λ Further, Upon using the constraint condition (7) we obtain
μ=
ξ
(15)
J•∂ξ /∂ J
We can then compute λ = 1/(μ − 1 ) if we so desire but it is not needed for subsequent considerations6 We thus get a normality rule with something beyond a geometrical interpretation, i.e., we are stating something about the physics of dissipative processes—that materials are such that among all the paths that they are allowed to choose, they will choose directions that locally maximizes the rate of dissipation7 3.3. Duality and the derived results Assuming that (14) is invertible, it is possible to “invert” the result (14) to obtain a Fenchel dual (Rockafellar, 2015) in the form
ˆ J=λ
∂φ ∂Q
(16)
where φ is the dual function to ξ and is a function of Q. Thus, for this case, normality in the space of “determinants” implies normality in the space of “resultants”. While mathematically the Eqs. (14) and (16) are equivalent under certain special conditions, they are not equivalent in physical interpretation. It is clear that (14) is the primary result in terms of maximum dissipation and that (16) is a derived result under further special assumptions. While one or the other is more convenient to use, there is a clear direct physical meaning to (14) whereas (16) is a derived result. This becomes evident when we consider the following: Since Q is a state vector that does not change at a particular instant, we can consider the following generalization: Given a rate of dissipation function ξ (Q, J) among all the choices for J that satisfy (7), the actual value for J is such that the rate of dissipation is maximized. Notice now that ξ is explicitly dependent on Q also, but since we are keeping Q constant and varying J we will get the same result (14) but now we will be unable to invert it to obtain (16) i.e. normality in the space of determinants does not imply normality in the space of resultants. For these cases, it is important to have a clear idea of that which is the cause and that which is the effect. 3.4. The Reiner–Rivlin Fluid For the case of the incompressible Reiner–Rivlin Fluid under homothermal conditions, we will assume that there is no “thermodynamic stress” so that J = σ = σ E and the resultants are Q = D where D = 1/2(∇ v + ∇ vt ) is the symmetric part of the velocity gradient. We shall assume that the rate of dissipation is a function only of the two principal invariants of deviatoric part (written as( · )dev of the stress, that is,
τ := σ dev ,
II := τ •τ ,
III := det(τ ),
ξ = ξˆ (II, III ),
(17)
Then, maximum rate of dissipation criterion (14) leads to
D=λ
∂ξ = λ(2ξ ,II τ + ξ ,III τ ∗dev ) ∂σ
(18)
where τ ∗ is the adjugate of τ and λ is given by
λ=
ξ ∂ξ /∂ τ •τ
(19)
The Eq. (18) is an implicit version of a Reiner–Rivlin Fluid8 and, in many cases can be inverted (by means of a Legendre– Fenchel transformation (Rockafellar, 2015) (see Rajagopal & Srinivasa, 2013 for details of its the application to dissipative The condition μ = 1, requires a different treatment see Rajagopal and Srinivasa (2005) and Srinivasa (2010). We point out that this variational principle does not generate the equations of motion but rather it determines a constitutive relation, and is similar to Gauss’ principle of least constraint in this regard. 8 Since (18) provides an explicit expression for D in terms of T, it is not strictly correct to call it an implicit relation. 6 7
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materials)) provided ξ is convex in its argument. We note that the assumption that ξ is only a function of the deviatoric stress guarantees isochoric response (i.e., incompressibility is a constitutive relation in this framework). Furthermore, the dependence of ξ as a function of III guarantees that there will be normal stress differences even for simple shear flow— that is, there is no problem in developing a Reiner–Rivlin Fluid that can exhibit normal stress differences in simple shear flow within the context of the maximum rate of dissipation criterion, contrary to statements in the work of Goddard (2014). 4. Extensions to elastoplasticity The extension to elastoplasticity requires careful consideration. As a first step the state variables need to be extended by including the plastic strain p as an additional variable and then (5) is extended to read:
σ E •∇ v + σ p •˙p − h•∇θ = θ ζ := ξ ≥ 0,
(20)
where σ p := −∂ u/∂ p . Now note that σ p •˙p is not in the form where the “determinant” and the “resultant” are obvious. So we have to find out what variable is held fixed and what can be varied before we decide upon the maximum dissipation criterion. To do this, we note that, unlike σ E , σ p is a function of the state variables and cannot be varied at a particular instant, rather ˙p is the variable that can be varied (since it is not constrained by any balance law)—in other words we should now require that ξ = ξ (σ E , h, ˙p ) and state the maximum rate of dissipation criterion in the form: Given the current state of the material, among all the choices of σ E , ˙p and h that are consistent with (20) which choice will maximize the rate of dissipation function ξ This approach will also provide a response function for the “resultants” in terms of the “determinants” in the form:
D=λ
∂ξ ∂ξ ∂ξ ,σ =λ , ∇θ = λ ∂ ˙p ∂h ∂ σE p
(21)
The value of λ will link the three responses unless we specifically split up the maximum rate of dissipation statement into individual functions for the separate physical phenomena. It is important to note that the normality rule for plasticity obtained in this way is the dual of the usual normality statement and the latter can be obtained under additional assumptions. However as shown by Srinivasa (2010), the form presented here actually is more versatile since it is able to include the responses for certain granular media (including the Coulomb–Mohr criteria) as well as for crushable foams (materials that do not satisfy the traditional normality conditions) into the same framework. In other words, it is important to understand the physical meaning of the variables that appear in the constitutive relations in terms of cause and effect and apply the maximum rate of dissipation criterion correctly in order to obtain meaningful results. Merely viewing them as mathematical or geometrical results can be misleading and obscures the critical role that physics plays in the elegant development of the appropriate mathematical approach. 5. Variational principles and certain aspects of the second law In view of the similarities between the classical thermodynamical results that the thermodynamic stresses are gradients of a potential, and the Eqs. (8) and (11), one can try and create a variational principle out of them. This is not a fruitful endeavor since the evolution equations for these variables are already assumed via the equations (2) without which the “determinants” and “resultants” cannot be defined. For the special case of linear relationships and that too only in the absence of convective effects as well as inertial effects, can one obtain a variational principle and interpret it as a principle of minimum rate of dissipation. This point has been discussed by Rajagopal and Srinivasa (2005) where they show how such a variational statement can be obtained. As has been noted in the case of the Gauss Principle of least constraint, these refer to rates and not to paths. Finally, a word of caution is necessary with regard to the usage of notion of the maximization of the rate of dissipation. As of now, it does not occupy the position of a universal law even within the context of classical continua and for the process classes that are considered. Rather, it is a highly restrictive assumption that may or may not be useful; it depends on the specific context. Even Gauss principle of minimum constraint is not universal as it is violated by constraints imposed by rough surfaces (i.e. frictional constraints). It is important to note that the original statements of the second law of thermodyanamics did not reference the notion of entropy production at all. As noted by Truesdell and Muncaster (1980) The objectives of continuum thermomechanics stop far short of explaining the “universe, but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size. Even in this form, we are only considering the total entropy of an isolated body. To go from here to the entropy production at any point in the body as being non-negative (much less that the process has to proceed in a way that the rate of entropy production be maximized) is quite a leap of faith and must be approached with abundant caution.
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Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ijengsci.2019.04. 003. References Carathéodory, C. (1909). Untersuchungen über die grundlagen der thermodynamik. Mathematische Annalen, 67(3), 355–386. Translation by J. Kestin appears in J. Kestin, The second law of thermodynamics, Benchmark papers on Energy/5, Dowden, Hutchinson and Ross, Inc., Stroudsburg, Pennsylvania (1976)) Carnot, S. (1960). Reflections on the motive power of fire, and on machines fitted to develop that power. Reflections on the Motive Power of Fire and Other Papers. New York: Dover Publications Inc, 3, 22. Clausius, R. (1862). On the application of the theorem of the equivalence of transformations to the internal work of a mass of matter. Philosophical Magazine, 81–97. Clausius, R. (1976). Different forms of the fundamental equations of the mechanical theory of heat and their convenience for application, in the second law of thermodynamics ed. J. Kestin (Dowden, Hutchinson and Ross, 1976), (translation of “Uber verschiedene fur die Anwendung bequeme Formen der Hauptgleichungen der mechnischen Warmetheorie, Abhandlungen uber die mechanische Warmethorie, Vol II, Vieweg, Braunschweig, 1–44 (1867)”. Edelen, D. (1973). On the existence of symmetry relations and dissipation potentials. Archive for Rational Mechanics and Analysis, 51(3), 218–227. Fosdick, R., & Rajagopal, K. (1983). On the existence of a manifold for temperature. Archive for Rational Mechanics and Analysis, 81(4), 317–332. Gauß, C. F. (1829). Über ein neues allgemeines grundgesetz der mechanik.. Journal für die reine und angewandte Mathematik, 4, 232–235. Glansdorff, P., & Prigogine, I. (1972). Thermodynamic theory of structure, stability and fluctuations. Wiley-Interscience, New York. Science, 176(4042), 1410–1420. Goddard, J. (2014). Edelen’s dissipation potentials and the visco-plasticity of particulate media. Acta Mechanica, 225(8), 2239–2259. Lord Rayleigh (1873). Some general theorems relating to vibration. Proceedings of the London Mathematical Society, 4, 366–368. Maxwell, J. (1891). Theory of heat. Longmans, Green. Onsager, L. (1931). Reciprocal relations in irreversible processes. i.. Physical Review, 37(4), 405–426. O’Reilly, O. M., & Srinivasa, A. R. (2001). On a decomposition of generalized constraint forces. Proceedings: Mathematics, Physical and Engineering Sciences, 1307–1313. Planck, M. (1949). Introduction to theoretical physics: Theory of heat: 5. Macmillan and Co. London. Rajagopal, K. (2011). Conspectus of concepts of elasticity. Mathematics and Mechanics of Solids, 16(5), 536–562. Rajagopal, K., & Srinivasa, A. (2005). On the nature of constraints for continua undergoing dissipative processes. In Proceedings of the royal society of london a: Mathematical, physical and engineering sciences: 461 (pp. 2785–2795). Rajagopal, K., & Srinivasa, A. (2008). On the thermodynamics of fluids defined by implicit constitutive relations. Zeitschrift für angewandte Mathematik und Physik, 59(4), 715–729. Rajagopal, K., & Srinivasa, A. (2013). An implicit thermomechanical theory based on a Gibbs potential formulation for describing the response of thermoviscoelastic solids. International Journal of Engineering Science, 70, 15–28. Rockafellar, R. (2015). Convex analysis. Princeton university press. Simpson, J., & Weiner, E. (20 0 0). The Oxford english dictionaryt (2nd). Oxford: Clarnendon Press, London. Srinivasa, A. (2010). Application of the maximum rate of dissipation criterion to dilatant, pressure dependent plasticity models. International Journal of Engineering Science, 48(11), 1590–1603. Stachel, J. (2016). The nature of the physical world: Gifford lectures of 1927: An annotated edition. Annotated with an introduction by H.G Callaway. xlix+ 381 pp., illus., bibl., index. Newcastle upon Tyne: Cambridge Scholars Publishing, 2014.. Truesdell, C. (2012). The elements of continuum mechanics. Lectures given in August-September 1965 for the Department of Mechanical and Aerospace Engineering Syracuse University Syracuse. New York: Springer Science & Business Media. Truesdell, C., & Muncaster, R. (1980). Fundamentals of Maxwel’s kinetic theory of a simple monatomic gas: Treated as a branch of rational mechanics: 83. Academic Press. Ziegler, H. (1963). Some extremum principles in irreversible thermodynamics, with application to continuum mechanics. Progress in Solid Mechanics, 4, 93–193. Ziegler, H. (1972). Systems with internal parameters obeying the orthogonality condition. Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 23(4), 553–566. Ziegler, H., & Wehrli, C. (1987). On a principle of maximal rate of entropy production. Journal of Non-Equilibrium Thermodynamics, 12(3), 229–244.
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International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Some remarks and clarifications concerning the restrictions placed on thermodynamic processes K.R. Rajagopal, A.R. Srinivasa∗ Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA
a r t i c l e
i n f o
Article history: Received 24 February 2019 Revised 25 April 2019 Accepted 26 April 2019
a b s t r a c t Various disparate restrictions are assumed with regard to bodies undergoing thermodynamic processes, in addition to the second law, to cull the class of constitutive relations for describing their response. In this paper we critically evaluate some of these additional restrictions. After a brief discussion of the various forms of the second law due to the pioneers in the field we turn our attention to an assessment of additional stringent restrictions due to Onsager, Prigogine, Ziegler, and Edelen. With the use of specific examples we show the pitfalls and inadequacies of these additional restrictions. The importance of identifying the “cause” (which we refer to as the “determinant” which is incorrectly referred to as “flux”) and its “effect” (which we refer to as the “resultant” which is incorrectly referred to as “affinity”) and not merely splitting the “determinant” on the basis of mathematical ease rather than understanding the underlying physics, is emphasized by considering specific examples. © 2019 Published by Elsevier Ltd.
1. Introduction This paper is devoted to a critical assessment of the validity and legitimacy of the disparate thermodynamic restrictions that are in vogue, that unfortunately do not speak with one voice. This dissonance is mostly a consequence of restrictions that are demanded in addition to the second law of thermodynamics, a law that is considered to be inviolate, which however is interpreted in a variety of ways. These additional restrictions are enforced to restrict the class of processes that a body can undergo as the second law allows for too large a class of processes to be possible, leading to a very large class of mathematical models from which one can make a choice for describing the response of bodies that one is interested in characterizing. While one might opine that having a very large class of constitutive relations to choose from could be viewed as being a very desirable situation, such models may exhibit other physically undesirable properties. Thus, it might be reasonable from the start to build in these additional requirements. Before we direct our attention to the confusion that is caused by the additional demands, it is necessary to discuss the various interpretations of the second law.The origins of the second law of thermodynamics can be traced to the seminal work of Carnot (1960) (Dover edition of the original which was published in 1824) entitled “Reflections on the Motive Power of Fire, and on Machines Fitted to Develop That Power”. Current versions of the second law cannot be easily recognized as stemming from that work, but nonetheless they do. There are numerous interpretations of the second law, starting with
∗
Corresponding author. E-mail address: [email protected] (A.R. Srinivasa).
https://doi.org/10.1016/j.ijengsci.2019.04.003 0020-7225/© 2019 Published by Elsevier Ltd.
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the following early statement due to Clausius1 (Clausius, 1976 also stated that: (1) The energy of the universe is constant, and (2) The entropy of the universe strives to attain a maximum), and Kelvin,2 and the inaccessibility statement due to Carathéodory (1909),3 the statement due to Planck (1949),4 the inequality that is referred to as the Clausius–Duhem inequality, etc. Irrespective of which statement is taken as the second law, the implications of all of them are the essentially the same, namely that heat cannot flow spontaneously from a colder to a hotter body and that the entropy of an isolated system cannot decrease or that the entropy of an isolated system tends to a maximum. The physicist Eddington Stachel (2016) bemoaned the inexorability and the cataclysmic consequences of the second law when he remarked “At present we can see no way in which an attack on the second law of thermodynamics could possibly succeed, and I confess personally I have no great desire that it should succeed in averting the final running-down of the universe.” While one comes across papers that discuss the possibility of the violation of the second law, supposedly at very small scales, these results are far from convincing and are yet to be accepted by the scientific community at large. This paper mainly considers the more stringent restrictions on thermodynamic processes than that required by the second law of thermodynamics. However, before getting into a discussion of these more exacting restrictions, a few remarks on how the second law itself is used as a constraint on thermodynamic processes of continua are in order. The second law as it stands concerns an isolated system and is a global statement. In continuum mechanics however the second law is assumed to hold for all arbitrary sub-parts of the body, which in the instance of all the variables in question being continuous leads to a local form of the law, and the second law is then enforced in the local form. This is a much more strict stipulation than the global form of the second law which allows the entropy in certain sub-parts within an isolated system to decrease as long as the entropy of the isolated system as a whole increases. One also finds the second law being used to enforce restrictions of thermodynamic processes associated with open systems which is clearly not the intent of the second law. One could argue that there are truly no closed systems that are sub-parts of our universe, every sub-part being in some way influenced by its environment and that the second law is an idealization and even as an idealization it might only be applicable to the Universe as a whole. Of course, the idealization that one could have such isolated bodies has stood us well with regard to explanations of processes in nature and as Eddington has observed, to date no such violation has been reported. Even if one accepts the use of the second law in its local form as a proper restriction on thermodynamic processes that a body undergoes, there is an additional requirement that is commonly demanded in continuum thermodynamics that is definitely not warranted, namely assuming that the body in question can undergo arbitrary thermodynamic processes, the second law in its local form (or to be more more precise, the constitutive relation that is used to describe the body), holding in all these arbitrary processes. The problem is a body cannot be described by the same constitutive relation in all arbitrary thermodynamic processes. In fact, the nature of a body is determined by the class of thermodynamic processes that it can undergo. Put differently, the class of constitutive relations that one assigns to a body is valid only for a certain class of processes. For instance, in the case of sufficiently small deformations that take place slowly, it might be reasonable to describe rubber as an elastic body. However, if the process is sufficiently fast, rubber can crystallize and one can no more consider is as elastic. Thus, in studying the response of a body to external stimuli, we need to know apriori the class of processes that the body can experience and find sufficient conditions, which might imply the class of constitutive relations that are appropriate, in order for the second law to be met. An even more telling example is that of a solid that melts. As a solid, it might have been described through constitutive relations for an elastic body and/or an inelastic body; as a liquid it would have a totally different constitutive relation. All that being said, let us assume that we can indeed enforce the second law of thermodynamics locally, dispense with our discussion of the second law and, and consider some of the stricter local restrictions that are demanded of thermodynamic processes. Onsager (1931)(see also Glansdorff and Prigogine (1972) for a discussion of the ideas of Onasager and the ensuing developments) using the assumption of microscopic reversibility, built upon the work of Lord Rayleigh (1873), and proposed a reciprocity principle that thermodynamic processes have to satisfy. He required that the rate of dissipation be minimized and showed that in an isolated body the reciprocity principle was equivalent to a variational statement that “the rate of increase in entropy, less the dissipation function, be a maximum”. Onsager’s reciprocity principle leads to linear laws such as Fourier’s law, Fick’s law, etc., and thus was hailed as a great advance in thermodynamics. Unfortunately, the scope of Onsager’s work is quite restrictive. Onsager’s reciprocal relations, and theorems related to it, only apply when the rate of entropy production is a quadratic form. The relations do not hold for yield-like phenomena exhibited by inelastic bodies and for most nonlinear response exhibited by bodies.
1 Clausius’ statement of the second law in Clausius (1862) as paraphrased by Thomson reads: “It is impossible for a self-acting machine, unaided by any external agency, to convey heat form one body to another at a higher temperature” 2 Kelvin’s statement of the second law [6] : “It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects”. 3 Caratheodory’s statement of the second law (Carathéodory, 1909): “In every arbitrarily close neighborhood of a given initial state there exists states which cannot be approached arbitrarily closely by adiabatic processes.” 4 Planck’s statement of the second law (Planck, 1949): “It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.”
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Ziegler (1963), Ziegler (1972) and Ziegler and Wehrli (1987) proposed two different “principles” wherein he required that rate of dissipation, expressed in terms of “Fluxes” and “Affinities” has to be orthogonal to the level surfaces of a dissipation function or that the rate of dissipation functions should be maximized. Rajagopal and Srinivasa (2008) also required that the rate of dissipation be maximized. However, as pointed out by them, there is a fundamental difference between the ideas of Ziegler (1963) and those of Rajagopal and Srinivasa (2008), in particular in the quantities that can be varied in the maximization process. In his work in 1963, Ziegler (1963) makes the following statements: “In a given stage of the deformation process, the entropy production inside the macrosystem is therefore a function of the (velocities) alone. So is the (rate of dissipation function) which is thus defined without recourse to the irreversible forces”. This statement is not true in general. The state variables that go into the determination of the rate of entropy production (rate of dissipation) are not merely velocities. Ziegler also makes the statement that (see page 134 line 3): “If the irreversible Force is prescribed, the actual quasistatic velocity maximizes the rate of dissipation”. That is, the maximization is achieved by keeping the force fixed and varying the velocities under the further assumption that the process is quasistatic. The focus on varying the velocities as opposed to varying the forces has deep philosophical implications related to the state of a dynamical system.In determining what quantities ought to be varied, one has to consider the appropriate dynamical system and determine its state variables so that “causes” and “effects” can be properly distinguished. This point is missed when considering classical thermodynamics since it is mostly concerned with quasistatic processes and cause and effect can be interchanged with suitable Legendre transformations. The use of the “generalized velocities” in cases where the dynamical system has inertia (so that the forces affect the accelerations) is philosophically problematic since, for such systems the position and velocity are both considered as part of description of the current state of the system and hence cannot be varied. What can be varied is the driving forces in these cases, since that is the cause for the change in the system. The situation is quite different for cases without inertia (for example with the plastic strain rate in classical plasticity) Ziegler’s approach, which focuses on generalized velocities alone will not lead to the determination of the equations that describe numerous phenomena. It will not lead to many of the implicit models that are used in mechanics. Practical utility aside, philosophically such an approach is wrong headed as it implies that the causes (namely forces) are kept constant and the velocities (effects) are maximized to obtain the results. Causality has been turned topsy-turvy! On the other hand, Rajagopal and Srinivasa (20 05, 20 08) require that “given the state variables of the dynamical system, the irreversible forces are such that the rate of dissipation is maximized”. That is, the effects (rate of change of the state variables) are the result of the causes (forces) which are changed to achieve the maximization. Ziegler’s approach can only lead to an expression for the stress in terms of the state variables and cannot yield implicit relationships between the forces and the state variables, this is clearly philosophically unsound as explained in detail by Rajagopal (2011). The shortcomings of Ziegler’s approach have been discussed in detail by Rajagopal and Srinivasa (2005) and Rajagopal and Srinivasa (2008). Other variational principles have been proposed (see. e.g., Edelen, 1973, Goddard (2014)) which have been adapted to study dissipative systems, but such approaches can be only used to study static problems since they explicitly ignore inertia and so they are inapplicable in dynamic problems(an example of the same is the recent work by Goddard (2014)). Such studies while applicable to the very special situation being considered might give the erroneous impression that they are valid when dynamics is to be taken into account. Given that one of the most important issues concerning the state of the body being its “stability”, dynamics cannot be ignored in most situations of interest. Furthermore, if one is truly interested in “thermodynamics” and not “thermostatics”, employing ideas that fail whenever dynamical effects come into play is clearly inappropriate. Rajagopal and Srinivasa (2005) have demonstrated how the ideas of Onsager could be made compatible with more general forms for the rate of entropy production and the requirement that the rate of entropy production be maximized without restrictions to quasistatic processes. There have been claims made in recent papers (see Goddard, 2014) that the proper way to establish constitutive relations for dissipative processes is to use the decomposition developed and used by Edelen (1973). In this paper we show that there is nothing particularly new to be gained by such an approach (since in these cases, it amounts to nothing more than a definition of a suitable vector and an explicit requirement that the terms orthogonal to it should be set to zero without any particular physical basis) and more importantly it cannot be applied to several situations where the requirement of the rate of entropy production should be maximum can be imposed, such as the dynamic problems mentioned earlier. Before getting into the main body of the work we would like to discuss a couple of issues that might seem nitpicky but which we think needs to be dealt with. The first is the poor choice of terminology that could confuse issues being discussed. In most of the papers devoted to irreversible thermodynamics, we come across the terms “fluxes” and “affinities” that are appended to certain variables. These terminologies are inappropriate to describe these quantities. What we have are quantities that can be identified as causes and those that can be identified as effects. For instance, consider the quantity that we refer to as stress. This is incorrectly labeled as being a flux. The word flux as a noun means flowing or flow and there is nothing about stress within the context of continuum mechanics (especially in solids) to think of it as flowing. The stress, which is a consequence of the contact force, causes deformation. As Truesdell (2012) remarks: “In continuum mechanics the forces of interest are contact forces, which are specified by the stress tensor.”; hence we shall refer to it as a “determinant”, an adjective that means that which determines the effect. The gradient of velocity in these works is identified as being the “Affinity” that corresponds to the stress. The word “Affinity means (Simpson & Weiner, 20 0 0) “an attractive force between substances and particles that causes them to enter into and remain in chemical combination”. There is nothing with regard
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to the nature of the velocity gradient to think of it as an “Affinity” in the sense of the above definition of an attractive force. If by “Affinity” one means “closely connected”, one cannot see any connection between the stress and the velocity gradient other than one being a cause and the other its effect. Thus, it would then make eminent sense to refer to the stress as a “determinant” and the velocity gradient as the “resultant”, that which is a result of the cause. Thus, we shall refer to quantities that cause to belong to the set of “determinants” and the effects to belong to the set of “resultants”. The second issue is the role of temperature, whether it is a “determinant” or a “resultant”. One comes across the following motivation/definition of temperature. According to the Simpson and Weiner (20 0 0), temperature is defined as “The state of a substance or body with regard to sensible warmth or coldness, referred to some standard of comparison, that quality or condition of a body which in degree varies directly with the amount of heat contained in the body”. And according to Maxwell (1891): “The temperature of a body is its thermal state considered with reference to the power of communicating heat to other bodies”. In both the above definitions, we notice that in order to define “temperature” the concept is based on first comprehending what is meant by “transfer of heat” as it is in the context of being able to transfer of heat that one understands the notion of “temperature”. Fosdick and Rajagopal (1983) have shown that by assuming the notion of the “transfer of heat” (transfer of thermal energy) one can prove that there exists a one-dimensional manifold that one can associate with the notion of “temperature”. Put differently, the concept of the notion of “transfer of heat” leads to the notion of “temperature”, thus we shall identify the “transfer of heat” as a “determinant” and the gradient of temperature as a “resultant”. The arrangement of the paper is as follows. After an introduction to the mathematical and thermodynamic preliminaries in Section 2, we discuss the procedures adopted for the determination of an entropy production (dissipation) potential and the status of Onsager’s reciprocity relation Onsager (1931), the work of Ziegler (1963), and the use of the ideas of Edelen (1973) by others Goddard (2014) and the consequences of requiring that such a choice be consistent with the demands of causality. We discuss the fact that Ziegler’s approach turns causality on its head and also does not allow one to obtain the cls of response functions that are necessary to describe a plethora of natural phenomena. Section 4 is devoted to a discussion of the procedure applied to the Reiner–Rivlin fluids and elasto-plastic solids. In the final section, we offer some cautionary remarks concerning variational generalizations of the second law. 2. Preliminaries Consider a body B that is made up of particles P that occupy positions X in an initial configuration κ r and x in the current configuration κ . The motion χ, the velocity v and the deformation gradient F are given by the usual definitions
x = χ ( X, t ),
v = x˙ ,
F=
∂x . ∂X
(1)
where the superposed dot represents the time derivative holding X fixed. For bodies that do not allow for diffusion, the motion is constrained by the conservation laws that take the form
˙ = −divv,
v˙ = divσ ,
u˙ = div(h ) + σ •L
(2)
where ϱ is the mass density, σ is the Cauchy stress and u is the specific internal energy, h is the heat flux and L := ∇ v is the velocity gradient. It is evident that these are in the form of a dynamical system whose evolution can be determined once constitutive relationships between the fluxes and the basic variables can be specified, so that the system can be closed. Specifically, once the initial conditions for x, v, ϱ and u are given, in principle the above Eqs. (1)b and (2 a,b,c) furnish sufficient equations for the evolution of the system. In determining the constitutive relations we have to begin with the fundamental equation of state which we introduce here in the form
η = ηˆ (F, , u ),
∂η 1 := ≥ 0 ∂u θ
(3)
which introduces the specific entropy η as a function of the internal energy and other kinematical variables, and whose evolution is governed by
η˙ = div(h/θ ) + ζ ,
ζ ≥ 0.
(4)
In the conventional approach to continuum thermodynamics, this additional equation is used as a constraint on the dynamical system, in the sense that the constitutive relations for the stress and the heat flux should be consistent with meeting the constraint for the class of processes under consideration. Here we caution the readers that the constraint ξ ≥ 0 is much more stringent than the statements of the second law of thermodynamics (which usually involve the whole body undergoing cycles of state). However, for our purpose we will retain the more stringent definition that we use here. The fundamental equation of state in the form (3) is too cumbersome to use and usually we use Legendre transforms to convert it to more convenient variables such as the Helmholtz or Gibbs Potentials. With the use of the Helmholtz potential, which can be written as ψ (F, ϱ, θ ), the consistency of the constitutive relations requires that η = −∂ψ /∂θ and
σ E •∇ v − h•∇θ = θ ζ , := ξ ≥ 0
(5)
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where the “extra stress” σ E is given by
∂ψ t 2 ∂ψ σ = σ − F + I , ∂F ∂ E
(6)
where the notation ( · )t represents the transpose operation. All that we need to do is to pick constitutive relations for σ and q such that (5) is non-negative for the process class under consideration. It is important to note that the statements of the second law of thermodynamics refer only to restrictions on the process classes (a typical statement being that due to Clausius (1862) that reads: “It is impossible for a self-acting machine, unaided by any external agency, to convey heat form one body to another at a higher temperature”.) and actually speaks nothing about constitutive relations. In keeping with this, if we choose any particular constitutive relation, then the non negativity of the rate of dissipation ξ will restrict the process class. 3. Further restrictions of the constitutive relations: Quest for a scalar potential for dissipative processes We note that Eq. (5) can be written as an inner product between the determinant variables J = {σ E , −h} and the resultants Q = {∇ v, ∇ θ } in the form
J •Q = ξ ≥ 0
(7)
The question is, what should we choose for Q? As with the case of hyperelastic response, there has been a concerted effort made to provide a rationale for obtaining Q in terms of a single scalar “potential”. Several geometrical as well as variational arguments have been made in the literature that are related to the dissipative response functions, the most notable of them being the Onsager reciprocity principle (which is based on the notion of microscopic reversibility) for linear constitutive relations and its related variational principle for steady state which is referred to as Prigogine’s principle of minimum rate of dissipation (Glansdorff & Prigogine, 1972). No such general result has been widely accepted in the literature with regard to non-linear dissipative response. The most widely used nonlinear dissipative response is provided by power law (or generalized Newtonian fluids) and elasto-plasticity. Formulation of the constitutive relations for the latter has traditionally pursued a geometric outlook using the notion of yield surfaces and loading and unloading directions. As has been extensively discussed by Rajagopal and Srinivasa (20 05, 20 08), Ziegler (1963, 1972) and Ziegler and Wehrli (1987) put forth two different, but related ideas based on the notion that not only the equation of state but a constitutive equation for the rate of dissipation in terms of the “resultants” Q needs to be provided and that (1) the “determinants” J should be such that the rate of dissipation be maximized. Later he modified it into a purely geometric criterion that the “determinants” should be along the normal to the surface of constant rate of dissipation. In a nutshell Ziegler (1972) approach can be stated in the form that the “determinants” J should be directed along the normal to level sets of the rate of dissipation function (Ziegler states this as being “orthogonal to the D-Surfaces”), i.e. since the normal to the level sets of ξ are given by ∂ ξ /∂ Q, Ziegler’s assumption implies that
ξ = ξˆ (Q ),
J=λ
∂ξ , ∂Q
(8)
the value of λ is determined by the condition (7). By substituting (8) into (7) and solving for λ we arrive at
λ=
ξ
(9)
Q•∂ξ /∂ Q
In a recent paper by Goddard (2014), this approach (and approaches related to this) was questioned—the argument being based on a purely mathematical decomposition of the flux vector J(q) (see Edelen, 1973). Specifically the approach advocated by Goddard (2014) is based on the following observation: Given any constitutive relation J(Q), (so that ξ (Q ) := J(Q )•(Q )) we can construct5 a function φ (Q) by setting
φ (q ) :=
1 0
1
λ
ξ (λQ ) dλ ⇒
∂φ • Q = ξ (Q ) ∂Q
(10)
In view of the above definition for φ , we can always decompose the vector J as
J=
∂φ + U, ∂Q
U •Q = 0
(11)
In other words, any “determinant” vector J which is a function of the “resultants” Q can be UNIQUELY written as a gradient of a scalar and a non-dissipative flux term U. They go on to posit that the non-dissipative term U be set to zero to describe purely dissipative systems.
5
The general result obtained by Edelen (1973) is vastly broader, but in the interest of clarity it is sufficient to discuss a more restricted version.
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Goddard points out cases where the results (11) (with U set to zero) and (8) are the same in many cases, and also that somehow (8) is incorrect since it is not generally in accordance with (11). He points to the case of Reiner–Rivlin fluids as an example. We show later that this claim is incorrect. As has been pointed out by Rajagopal and Srinivasa (20 05, 20 08), there are several issues with both the approaches (11) and (8) none of them having to do with the mathematics of the approaches but with their philosophy. We first emphasize that, irrespective of the philosopical issues, the assumption U = 0 in (11) is a completely ad-hoc constitutive choice that is not based on any physical notions but a purely mathematical assumption. On the other hand, (8) is based on a physical assumption. 3.1. Cause, effect and teleology Classical thermodynamics, unlike Quantum thermodynamics, is well anchored in determinism and causality plays a central role in its development. Delving deeply into the philosophical underpinning of classical thermodynamics is not necessary in making our case with regard to the issues on hand. A cursory understanding of the role of causality is sufficient for our purposes. In the particular case of dissipative response, from a purely mathematical point of view, one can establish the following general result: Given ξ (Q) and any scalar valued function φ (Q) whose level sets are star shaped with respect to the origin, it is always possible to write the “determinants” J as
J=λ
∂φ + U, ∂Q
U •Q = 0
(12)
The proof is trivial: simply set λ = ξ (Q )/(Q•∂φ /∂ Q ). The denominator is non zero owing to the star shaped nature of the level sets and the result follows. It is now possible to choose U = 0 and then for any function ξ we will have a constitutive relation in terms of a scalar function φ . In the context of plasticity, these two functions ξ and φ are related to the yield function and the loading function, respectively. The two special cases are when this function is ξ itself, in which case we get the orthogonality condition of Ziegler and when this function is defined by (10) we get (11). Again, within the context of plasticity, the identification of the yield and loading functions as being the same gives rise to the “normality rule”. Thus, mathematically there is nothing particularly unique in preferring Ziegler’s orthogonality (Ziegler & Wehrli, 1987) or (11), other than the requirement of specifying two functions instead of one. A similar situation is found with constraint responses, where we have the geometrical idea that the generalized constraint force is normal to the constraint surface. A search for a physical reason for this condition led to two ideas–that the constraints did no virtual work and further the “Gauss principle of least constraint” (Gauß (1829), See also O’Reilly and Srinivasa (2001) and Rajagopal and Srinivasa (2005) for applications in continua), which provides a physical basis for the assumed normality condition. This points to the need for seeking other physical explanations for why one might prefer one version versus the other. We propose to offer such an explanation here. 3.2. Cause and effect are reversed in the conventional approaches of Ziegler and Wehrli (1987) and Goddard (2014) We first note that an “orthogonality condition” such as (11) has no “meaning” other than geometric convenience which may be corroborated (or at least not completely disproved) by experimental evidence. In general for it to have physical meaning, such an orthorogonality condition should be connected to a statement of physical significance. For example, the “normality” rule of plasticity is connected with the notion of maximum plastic work or the idea of the work inequality both of which provide a physical justification for the geometrical idea. When we consider such an approach, we also need to consider: In order to describe an optimum principle, we need to know the parameters that are fixedby the current state of the system and the parameters that can be varied. We note that as a dynamical system given by (2), the causes that drive the system are the “determinants”, i.e. J is the cause. Moreover, at any instant of time Q is already specified. Thus, any maximum principle that is generally valid should consider the (as yet unspecified) “determinants” J as being varied and not the “resultants” Q. In terms of meaning, if we assume that ξ is a function of Q, its value is fixed at each instant of time (since the state is known). So it is not suitable to consider ξ as a function only of the state variables, it should be considered as a function of the as yet unspecified causal variables J so that its values can be varied by choosing different values for J. In other words for the dynamical system considered above, it is proper to consider a maximum dissipation assumption of the form: Given a rate of dissipation function ξ (J) among all the choices for J that satisfy (7) for a given Q), the actual values for J are such that the rate of dissipation is maximized. The above problem can be stated as a constrained maximization of a function f (λ, J ) = (ξ (J ) − λ(Q•J − ξ (J )) ) (using a Lagrange multiplier to enforce the constraint (7)
J∗ (Q ) = argmaxJ f (λ, J )
(13)
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Under suitable smoothness assumptions on ξ (J), so that the standard methods of calculus can be applied, the above equation implies that the resultants are normal to the constant dissipation surface in “determinant” space, i.e.,
∂f ∂ξ =0⇒Q=μ , ∂J ∂J
(14)
where μ = (1 + λ )/λ Further, Upon using the constraint condition (7) we obtain
μ=
ξ
(15)
J•∂ξ /∂ J
We can then compute λ = 1/(μ − 1 ) if we so desire but it is not needed for subsequent considerations6 We thus get a normality rule with something beyond a geometrical interpretation, i.e., we are stating something about the physics of dissipative processes—that materials are such that among all the paths that they are allowed to choose, they will choose directions that locally maximizes the rate of dissipation7 3.3. Duality and the derived results Assuming that (14) is invertible, it is possible to “invert” the result (14) to obtain a Fenchel dual (Rockafellar, 2015) in the form
ˆ J=λ
∂φ ∂Q
(16)
where φ is the dual function to ξ and is a function of Q. Thus, for this case, normality in the space of “determinants” implies normality in the space of “resultants”. While mathematically the Eqs. (14) and (16) are equivalent under certain special conditions, they are not equivalent in physical interpretation. It is clear that (14) is the primary result in terms of maximum dissipation and that (16) is a derived result under further special assumptions. While one or the other is more convenient to use, there is a clear direct physical meaning to (14) whereas (16) is a derived result. This becomes evident when we consider the following: Since Q is a state vector that does not change at a particular instant, we can consider the following generalization: Given a rate of dissipation function ξ (Q, J) among all the choices for J that satisfy (7), the actual value for J is such that the rate of dissipation is maximized. Notice now that ξ is explicitly dependent on Q also, but since we are keeping Q constant and varying J we will get the same result (14) but now we will be unable to invert it to obtain (16) i.e. normality in the space of determinants does not imply normality in the space of resultants. For these cases, it is important to have a clear idea of that which is the cause and that which is the effect. 3.4. The Reiner–Rivlin Fluid For the case of the incompressible Reiner–Rivlin Fluid under homothermal conditions, we will assume that there is no “thermodynamic stress” so that J = σ = σ E and the resultants are Q = D where D = 1/2(∇ v + ∇ vt ) is the symmetric part of the velocity gradient. We shall assume that the rate of dissipation is a function only of the two principal invariants of deviatoric part (written as( · )dev of the stress, that is,
τ := σ dev ,
II := τ •τ ,
III := det(τ ),
ξ = ξˆ (II, III ),
(17)
Then, maximum rate of dissipation criterion (14) leads to
D=λ
∂ξ = λ(2ξ ,II τ + ξ ,III τ ∗dev ) ∂σ
(18)
where τ ∗ is the adjugate of τ and λ is given by
λ=
ξ ∂ξ /∂ τ •τ
(19)
The Eq. (18) is an implicit version of a Reiner–Rivlin Fluid8 and, in many cases can be inverted (by means of a Legendre– Fenchel transformation (Rockafellar, 2015) (see Rajagopal & Srinivasa, 2013 for details of its the application to dissipative The condition μ = 1, requires a different treatment see Rajagopal and Srinivasa (2005) and Srinivasa (2010). We point out that this variational principle does not generate the equations of motion but rather it determines a constitutive relation, and is similar to Gauss’ principle of least constraint in this regard. 8 Since (18) provides an explicit expression for D in terms of T, it is not strictly correct to call it an implicit relation. 6 7
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materials)) provided ξ is convex in its argument. We note that the assumption that ξ is only a function of the deviatoric stress guarantees isochoric response (i.e., incompressibility is a constitutive relation in this framework). Furthermore, the dependence of ξ as a function of III guarantees that there will be normal stress differences even for simple shear flow— that is, there is no problem in developing a Reiner–Rivlin Fluid that can exhibit normal stress differences in simple shear flow within the context of the maximum rate of dissipation criterion, contrary to statements in the work of Goddard (2014). 4. Extensions to elastoplasticity The extension to elastoplasticity requires careful consideration. As a first step the state variables need to be extended by including the plastic strain p as an additional variable and then (5) is extended to read:
σ E •∇ v + σ p •˙p − h•∇θ = θ ζ := ξ ≥ 0,
(20)
where σ p := −∂ u/∂ p . Now note that σ p •˙p is not in the form where the “determinant” and the “resultant” are obvious. So we have to find out what variable is held fixed and what can be varied before we decide upon the maximum dissipation criterion. To do this, we note that, unlike σ E , σ p is a function of the state variables and cannot be varied at a particular instant, rather ˙p is the variable that can be varied (since it is not constrained by any balance law)—in other words we should now require that ξ = ξ (σ E , h, ˙p ) and state the maximum rate of dissipation criterion in the form: Given the current state of the material, among all the choices of σ E , ˙p and h that are consistent with (20) which choice will maximize the rate of dissipation function ξ This approach will also provide a response function for the “resultants” in terms of the “determinants” in the form:
D=λ
∂ξ ∂ξ ∂ξ ,σ =λ , ∇θ = λ ∂ ˙p ∂h ∂ σE p
(21)
The value of λ will link the three responses unless we specifically split up the maximum rate of dissipation statement into individual functions for the separate physical phenomena. It is important to note that the normality rule for plasticity obtained in this way is the dual of the usual normality statement and the latter can be obtained under additional assumptions. However as shown by Srinivasa (2010), the form presented here actually is more versatile since it is able to include the responses for certain granular media (including the Coulomb–Mohr criteria) as well as for crushable foams (materials that do not satisfy the traditional normality conditions) into the same framework. In other words, it is important to understand the physical meaning of the variables that appear in the constitutive relations in terms of cause and effect and apply the maximum rate of dissipation criterion correctly in order to obtain meaningful results. Merely viewing them as mathematical or geometrical results can be misleading and obscures the critical role that physics plays in the elegant development of the appropriate mathematical approach. 5. Variational principles and certain aspects of the second law In view of the similarities between the classical thermodynamical results that the thermodynamic stresses are gradients of a potential, and the Eqs. (8) and (11), one can try and create a variational principle out of them. This is not a fruitful endeavor since the evolution equations for these variables are already assumed via the equations (2) without which the “determinants” and “resultants” cannot be defined. For the special case of linear relationships and that too only in the absence of convective effects as well as inertial effects, can one obtain a variational principle and interpret it as a principle of minimum rate of dissipation. This point has been discussed by Rajagopal and Srinivasa (2005) where they show how such a variational statement can be obtained. As has been noted in the case of the Gauss Principle of least constraint, these refer to rates and not to paths. Finally, a word of caution is necessary with regard to the usage of notion of the maximization of the rate of dissipation. As of now, it does not occupy the position of a universal law even within the context of classical continua and for the process classes that are considered. Rather, it is a highly restrictive assumption that may or may not be useful; it depends on the specific context. Even Gauss principle of minimum constraint is not universal as it is violated by constraints imposed by rough surfaces (i.e. frictional constraints). It is important to note that the original statements of the second law of thermodyanamics did not reference the notion of entropy production at all. As noted by Truesdell and Muncaster (1980) The objectives of continuum thermomechanics stop far short of explaining the “universe, but within that theory we may easily derive an explicit statement in some ways reminiscent of Clausius, but referring only to a modest object: an isolated body of finite size. Even in this form, we are only considering the total entropy of an isolated body. To go from here to the entropy production at any point in the body as being non-negative (much less that the process has to proceed in a way that the rate of entropy production be maximized) is quite a leap of faith and must be approached with abundant caution.
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